A Global Theory on Variational Approximations of Linear Problems: Some Remarkable Applications

Abstract

In the early seventies Babushka 3 set up the bases of a general approximation theory of linear variational problems. Over ten years later, Dupire 6 refined this theory in his doctoral thesis defended at PUC-Rio, the Catholic University of Rio de Janeiro. This together with classical estimates of the polynomial interpolation error in Sobolev norms, has since been widely used as the basic tool to establish the convergence of finite element solutions of partial differential equations. The purpose of this work is two-fold: First we endeavour to recall the main results of Dupire6 , while pointing out some of its yet unexploited aspects; Then we show through a simple example, how both ingredients allow a straightforward convergence analysis of the finite volume method as well.